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Quantum Chaos

  • Reference work entry
Encyclopedia of Complexity and Systems Science

Definition of the Subject

As it is now widely recognized, classical dynamical chaos has been one of the major scientific breakthroughs of the past century. Quantum chaos ,sometimes called Quantum chaology, studies the manifestations of chaotic motion and related dynamical phenomena in quantum mechanics [1,2].

More abstractly, one may define as quantum chaos those phenomena of simple quantum systems which can be describedstatistically and exhibit some universal (i. e. system independent) features. By the term simple we mean herethat the system can be specified by a finite set of parameters or, generally, can be described by a finite amount of information. So we canfundamentally distinguish the phenomena of quantum chaos from similar dynamical phenomena in disordered systems –specified in terms of random parameters and which therefore contain infinite amount of information in an appropriate (say thermodynamic) limit.

The universal statistical properties of quantum chaotic systems which...

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Abbreviations

Ergodicity :

The property of a dynamical system, according to which a single trajectory, starting from almost any initial condition, explores (densely covers) the entire available phase space of physical states.

Integrability :

A classical Hamiltonian dynamical system of N degrees of freedom is said to be integrable (according to Liouville) if there exist N independent conserved quantities. Integrability implies explicit quasi-periodic solution of the equations of motion.

Random matrix theory :

The statistical theory which allows to describe the fluctuation properties of quantum systems in terms of the sets (ensembles) of random Hermitian matrices with appropriate invariant measures.

Wigner surmise :

Nearest neighbor energy level spacing distribution based on the simplest \( { 2\times 2 } \) Gaussian Hermitian random matrix models, accurately approximating spacing distributions in complex quantum systems.

Periodic orbit theory or trace formula :

A relationship between certain properties of energy spectrum of a quantized chaotic system, and the set of unstable periodic orbits of the corresponding classical chaotic system.

Quantum Loschmidt echo or fidelity :

A measure of stability of quantum time evolution. It is computed as a Hilbert space inner product of two slightly different quantum time evolutions starting from the same initial state.

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Author information

Authors and Affiliations

  1. Universita dell'Insubria, Como, Italy

    Giulio Casati

  2. Univerza v Ljubljani, Ljubljana, Slovenia

    Tomaž Prosen

Authors
  1. Giulio Casati
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  2. Tomaž Prosen
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Editor information

Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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© 2009 Springer-Verlag

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Casati, G., Prosen, T. (2009). Quantum Chaos. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_427

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